Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in

The Production Function - Microeconomics

  1. A firm uses two inputs, skilled labor (S) and unskilled labor (U). Its production function is Q=10S+5UQ = 10S + 5U. The firm currently employs 20 skilled workers and 50 unskilled workers. If the firm must reduce its skilled labor employment by 2 workers, how many additional unskilled workers must it hire to keep its output constant?
    Answer: 4 workers First, calculate the current output: Q = 10(20) + 5(50) = 200 + 250 = 450. Next, calculate the output with the reduced skilled labor: S' = 20 - 2 = 18. The output from skilled labor is now 10(18) = 180. To keep total output at 450, the output from unskilled labor must be 450 - 180 = 270. The contribution per unskilled worker is 5 units. To get 270 units of output, the firm needs 270 / 5 = 54 unskilled workers. The firm started with 50 unskilled workers, so it must hire 54 - 50 = 4 additional unskilled workers. Alternatively, one can use the MRTS. MP_S = 10, MP_U = 5. MRTS_SU = MP_S/MP_U = 10/5 = 2. This means 1 unit of S is equivalent to 2 units of U. To replace 2 units of S, the firm needs 2 * 2 = 4 units of U.
  2. A production function is given by Q=ALαKβQ = AL^{\alpha}K^{\beta} where A > 0, and both α and β are positive constants. If the elasticity of output with respect to labor is 0.4 and the elasticity of output with respect to capital is 0.6, what can be concluded about this production function's returns to scale and the relationship between average and marginal products?
    Answer: The function exhibits constant returns to scale, and marginal products equal average products at all input levels When you encounter a Cobb-Douglas production function like Q=ALαKβQ = AL^{\alpha}K^{\beta}, you're looking at a relationship where the exponents α and β tell you crucial information about both elasticities and returns to scale. The elasticity of output with respect to each input equals its exponent in the Cobb-Douglas function. Since the elasticity with respect to labor is 0.4, we know α = 0.4. Since the elasticity with respect to capital is 0.6, we know β = 0.6. This gives us Q=AL0.4K0.6Q = AL^{0.4}K^{0.6}. Returns to scale are determined by adding the exponents: α + β = 0.4 + 0.6 = 1.0. When this sum equals 1, the function exhibits constant returns to scale—doubling all inputs exactly doubles output. For Cobb-Douglas functions with constant returns to scale, there's a special property: marginal products always equal average products. This occurs because the function is homogeneous of degree 1, creating a proportional relationship between total, average, and marginal products at every input level. Answer A incorrectly suggests decreasing returns to scale (which would require α + β < 1) and wrong product relationships. Answer B correctly identifies constant returns to scale but wrongly states that marginal products are less than average products—this relationship doesn't hold for constant returns to scale Cobb-Douglas functions. Answer C incorrectly suggests increasing returns to scale (requiring α + β > 1). Remember: In Cobb-Douglas functions, sum the exponents to determine returns to scale, and with constant returns to scale, marginal always equals average product.
  3. Consider two firms with production functions QA=L+2KQ_A = L + 2K and QB=L0.5K0.5Q_B = L^{0.5}K^{0.5}. Which of the following statements correctly identifies a key distinction between their production technologies?
    Answer: Firm B's technology is subject to diminishing marginal returns, while Firm A's is not. Let's analyze the marginal products (MP). For Firm A (perfect substitutes): MP_L = ∂Q/∂L = 1 and MP_K = ∂Q/∂K = 2. The marginal products are constant, so Firm A's technology does not exhibit diminishing marginal returns. For Firm B (Cobb-Douglas): MP_L = 0.5L⁻⁰.⁵K⁰.⁵. As L increases (holding K constant), MP_L decreases. Thus, Firm B's technology is subject to diminishing marginal returns. Distractor A is false because both firms exhibit constant returns to scale. Distractor C is false because Firm A can produce with one input, while Firm B cannot. Distractor D is false because Firm A has linear isoquants and Firm B has convex isoquants.
  4. A delivery service requires exactly one driver (L) for each van (K). Hiring an additional driver without an additional van does not increase the number of deliveries, nor does acquiring an extra van without an extra driver. Which statement accurately describes the isoquants for this production process?
    Answer: The isoquants are L-shaped, with the vertex occurring where the ratio of capital to labor is 1:1. This scenario describes a Leontief or fixed-proportions production function, as the inputs (drivers and vans) are perfect complements. The production function is Q = min(L, K). For any given output level, say Q=10, the firm must use L=10 and K=10. Adding more L (e.g., L=11, K=10) or more K (e.g., L=10, K=11) does not increase output. This creates L-shaped isoquants, with the vertex (corner) occurring where the inputs are in the required 1:1 ratio.
  5. A firm's production function is initially Q1=10L0.5K0.5Q_1 = 10 L^{0.5}K^{0.5}. A neutral technological improvement occurs, resulting in the new production function Q2=12L0.5K0.5Q_2 = 12 L^{0.5}K^{0.5}. How does this change affect the marginal product of labor (MPLMP_L) and the marginal rate of technical substitution (MRTSLKMRTS_{LK}) at any given combination of L and K?
    Answer: MPLMP_L increases, and MRTSLKMRTS_{LK} remains constant. The technological improvement is represented by an increase in the multiplicative factor from 10 to 12. 1. The original marginal product of labor is MP_L1 = ∂Q₁/∂L = 5L⁻⁰.⁵K⁰.⁵. The new one is MP_L2 = ∂Q₂/∂L = 6L⁻⁰.⁵K⁰.⁵. Since 6 > 5, MP_L increases. 2. The MRTS_LK is the ratio MP_L / MP_K. The original MP_K1 = 5L⁰.⁵K⁻⁰.⁵, so MRTS₁ = (5L⁻⁰.⁵K⁰.⁵)/(5L⁰.⁵K⁻⁰.⁵) = K/L. The new MP_K2 = 6L⁰.⁵K⁻⁰.⁵, so MRTS₂ = (6L⁻⁰.⁵K⁰.⁵)/(6L⁰.⁵K⁻⁰.⁵) = K/L. The MRTS remains unchanged. This is a characteristic of neutral technological change in a Cobb-Douglas function.
  6. A firm's production function is Q=f(L,K)Q = f(L,K) where the marginal product of labor is MPL=20−2LMP_L = 20 - 2L and the marginal product of capital is MPK=30−3KMP_K = 30 - 3K. If the firm currently uses 6 units of labor and 8 units of capital, what is the rate of technical substitution of capital for labor (MRTS) at this input combination?
    Answer: The MRTS equals 1.33, indicating that 1.33 units of capital can substitute for 1 unit of labor When you encounter questions about the marginal rate of technical substitution (MRTS), you're dealing with how firms can substitute one input for another while maintaining the same output level. The MRTS of capital for labor tells you how many units of capital you need to give up to use one additional unit of labor. The formula for MRTS of capital for labor is: MRTS=MPLMPKMRTS = \frac{MP_L}{MP_K} First, calculate the marginal products at the given input levels (L = 6, K = 8): - MPL=20−2(6)=20−12=8MP_L = 20 - 2(6) = 20 - 12 = 8 - MPK=30−3(8)=30−24=6MP_K = 30 - 3(8) = 30 - 24 = 6 Therefore: MRTS=86=1.33MRTS = \frac{8}{6} = 1.33 This means 1.33 units of capital can substitute for 1 unit of labor, making answer D correct. Answer A (0.67) incorrectly inverts the formula, calculating MPKMPL\frac{MP_K}{MP_L} instead of MPLMPK\frac{MP_L}{MP_K}. Answer B (0.75) makes the same inversion error but with an additional calculation mistake. Answer C (1.50) appears to use incorrect marginal product values, possibly confusing the input levels or making arithmetic errors. Study tip: Always remember that MRTS of capital for labor equals MPLMPK\frac{MP_L}{MP_K}, not the reverse. The marginal product of the input you're adding more of (labor) goes in the numerator, and the marginal product of the input you're reducing (capital) goes in the denominator.
  7. The manager of a call center with 20 agents observes that the team's average output is 15 resolved calls per agent per hour. The manager adds a 21st agent, and total team output increases to 310 resolved calls per hour. What is the effect of hiring the 21st agent on the average product of labor (APLAP_L), and what does this imply about the marginal product (MPLMP_L) of that agent?
    Answer: APLAP_L decreases because the MPLMP_L of the 21st agent is below the previous average product. 1. Calculate the initial total product: 20 agents * 15 calls/agent = 300 calls. 2. The new total product with 21 agents is 310 calls. 3. The marginal product of the 21st agent is the change in total product: 310 - 300 = 10 calls. 4. The initial average product was 15 calls. Since the marginal product of the new agent (10) is less than the previous average product (15), the new average product must decrease. 5. We can verify the new average product: 310 calls / 21 agents ≈ 14.76 calls/agent, which is a decrease from 15.
  8. A consulting firm produces reports (Q) using analysts (L) and computers (K). The production function is Q=L2KQ=L^2K. In the short run, the number of computers is fixed at K=4. What is the marginal product of the 3rd analyst hired?
    Answer: 20 The marginal product of an input is the additional output produced by one more unit of that input. Here, we want the marginal product of the 3rd analyst, which is the total output with 3 analysts minus the total output with 2 analysts. 1. First, write the short-run production function by substituting K=4: Q = L²(4) = 4L². 2. Calculate total output with L=2 analysts: Q(2) = 4(2²) = 4(4) = 16. 3. Calculate total output with L=3 analysts: Q(3) = 4(3²) = 4(9) = 36. 4. The marginal product of the 3rd analyst is the difference: MP₃ = Q(3) - Q(2) = 36 - 16 = 20. Alternatively, using calculus, MP_L = dQ/dL = 8L. This is the instantaneous marginal product. The question asks for the discrete change. A common distractor would be to use the derivative at L=3 (MP=24) or L=2 (MP=16). Using the average of the two points is a better approximation: MP at L=2.5 is 8(2.5)=20.
  9. Consider a firm with production function Q=L0.6K0.4Q = L^{0.6}K^{0.4}. The firm faces input prices of $20 per unit of labor and $40 per unit of capital. If the firm wants to produce exactly 100 units of output at minimum cost, what is the optimal ratio of capital to labor (K/L) that the firm should employ?
    Answer: The optimal K/L ratio is 0.67 based on the input price ratio and output elasticities For cost minimization, the firm sets MRTS = w/r, where MRTS = MPL/MPK. For this Cobb-Douglas function: MPL = 0.6L−0.40.6L^{-0.4}K0.4K^{0.4} and MPK = 0.4L^0.6K^(-0.6). So MRTS = (0.6/0.4)(K/L) = 1.5(K/L). Setting equal to price ratio: 1.5(K/L) = 20/40 = 0.5. Solving: K/L = 0.5/1.5 = 1/3 ≈ 0.67. Choice A uses incorrect MRTS calculation. Choice B inverts the optimal ratio. Choice D makes an error in combining exponents and prices.
  10. A firm's production process is described by the function Q=L0.5K0.25M0.25Q=L^{0.5}K^{0.25}M^{0.25}, where L is labor, K is capital, and M is raw materials. What are the returns to scale for this production function?
    Answer: Constant returns to scale This is a multi-input Cobb-Douglas production function. The returns to scale are determined by the sum of the exponents of all inputs. In this case, the sum is 0.5 + 0.25 + 0.25 = 1.0. When the sum of the exponents equals 1, the production function exhibits constant returns to scale. This means that if all inputs (L, K, and M) are doubled, the output (Q) will also double.
  11. A production function is given by Q=100L0.3K0.8Q = 100L^{0.3}K^{0.8}. Which of the following statements accurately describes this production function?
    Answer: It exhibits diminishing marginal returns to labor and increasing returns to scale. This is a Cobb-Douglas production function. 1. Marginal returns to a factor: The exponent of labor (L) is 0.3, which is less than 1. This indicates diminishing marginal returns to labor (as L increases, holding K constant, its marginal product will decrease). 2. Returns to scale: This is determined by the sum of the exponents of the inputs. The sum is 0.3 + 0.8 = 1.1. Since the sum is greater than 1, the production function exhibits increasing returns to scale (doubling all inputs will more than double output).
  12. A firm operates with the production function Q=min{4L,2K}Q = min\{4L, 2K\} where L is labor hours and K is machine hours. If the firm currently produces 80 units of output and operates at the optimal input combination, what happens to total output if the firm increases labor by 10 hours while keeping capital constant?
    Answer: Output remains at 80 units because capital becomes the binding constraint This is a Leontief (fixed proportions) production function. At the optimum, 4L = 2K, so L = K/2. With Q = 80, we have 4L = 80, so L = 20 and K = 40. The current input ratio is optimal. Adding 10 hours of labor gives L = 30, but K remains 40. Now Q = min{4(30), 2(40)} = min{120, 80} = 80. Capital becomes the binding constraint, so output doesn't change. Choice A ignores the constraint nature. Choice C misunderstands fixed proportions. Choice D incorrectly applies marginal product concepts to Leontief functions.
  13. A manufacturer observes that when it hires more machine operators to work on its existing stock of equipment, each additional operator adds less to total output than the previous one. Based solely on this information, what can be concluded about the firm's long-run production technology?
    Answer: The information is insufficient to determine the nature of returns to scale. The scenario describes diminishing marginal returns to labor, which is a short-run phenomenon that occurs when one input (labor) is increased while another (capital) is held fixed. Returns to scale is a long-run concept describing the change in output when all inputs are increased proportionally. A production function can exhibit diminishing marginal returns to a single factor while simultaneously having increasing, decreasing, or constant returns to scale. Therefore, the information provided is not sufficient to make a conclusion about returns to scale.
  14. A firm's production function is Q=K+LQ = K + \sqrt{L}. The firm is currently using K=10 units of capital and L=16 units of labor. What is the marginal rate of technical substitution (MRTSLKMRTS_{LK}) at this point?
    Answer: 1/8 The marginal rate of technical substitution (MRTS_LK) is the ratio of the marginal products, MP_L / MP_K. 1. Find the marginal product of labor: MP_L = ∂Q/∂L = (1/2)L⁻¹/² = 1/(2√L). 2. Find the marginal product of capital: MP_K = ∂Q/∂K = 1. 3. Form the ratio for MRTS_LK: MRTS_LK = MP_L / MP_K = [1/(2√L)] / 1 = 1/(2√L). 4. Substitute the current level of labor, L=16: MRTS_LK = 1/(2√16) = 1/(2*4) = 1/8. Notice that for this particular production function, the MRTS does not depend on the amount of capital K.
  15. A firm's short-run production function is given by Q=−2L3+24L2+120LQ = -2L^3 + 24L^2 + 120L, where L is the number of workers. The economically rational range of production (Stage II) begins where the average product of labor is maximized and ends where total product is maximized. What is the size of this range of labor employment?
    Answer: 4 units of labor Stage II begins at the maximum of the average product of labor (APLAP_L) and ends at the maximum of total product (TP), where the marginal product of labor (MPLMP_L) is zero. 1. Find the start of Stage II: AP_L = Q/L = -2L² + 24L + 120. To find its maximum, set its derivative to zero: d(APLAP_L)/dL = -4L + 24 = 0, which gives L = 6. 2. Find the end of Stage II: TP is maximized when MP_L = 0. MP_L = dQ/dL = -6L² + 48L + 120. Setting this to zero: -6(L² - 8L - 20) = 0. Factoring gives -6(L-10)(L+2) = 0. The economically relevant solution is L = 10. 3. The size of the range is the difference between the end and the start: 10 - 6 = 4 units of labor.
  16. A production function exhibits the property that f(tL,tK)=t1.2f(L,K)f(tL, tK) = t^{1.2}f(L,K) for any positive scalar t. If this firm increases all inputs by 25%, what is the resulting percentage change in total output, and what does this imply about the firm's long-run average cost?
    Answer: Output increases by 44.14%, implying decreasing long-run average cost due to increasing returns to scale The function exhibits increasing returns to scale with degree 1.2. When inputs increase by 25% (t = 1.25): new output = (1.25)1.2(1.25)^{1.2} × original output = 1.4414 × original output. This represents a 44.14% increase in output. Since output increases more than proportionally to input increases, the firm has increasing returns to scale, which implies decreasing long-run average cost (economies of scale). Choice A assumes constant returns. Choice B incorrectly calculates 1.25² = 1.5625. Choice D assumes decreasing returns to scale.
  17. A firm's production function is given by Q=L1/3K2/3Q = L^{1/3}K^{2/3}. If the firm is currently using 3 units of labor (L) and 18 units of capital (K), what is the marginal rate of technical substitution of labor for capital (MRTSLKMRTS_{LK})?
    Answer: 3 The marginal rate of technical substitution (MRTS_LK) is the ratio of the marginal products, MP_L / MP_K. 1. Calculate the marginal product of labor: MP_L = ∂Q/∂L = (1/3)L⁻²/³K²/³. 2. Calculate the marginal product of capital: MP_K = ∂Q/∂K = (2/3)L¹/³K⁻¹/³. 3. Form the ratio: MRTS_LK = MP_L / MP_K = [ (1/3)L⁻²/³K²/³ ] / [ (2/3)L¹/³K⁻¹/³ ] = (1/2)(K/L). 4. Substitute the given values L=3 and K=18: MRTS_LK = (1/2)(18/3) = (1/2)(6) = 3.
  18. A firm's production function is Q=min⁡(L,3K)Q = \min(L, 3K). In the short run, capital is fixed at K=5. Which expression represents the marginal product of labor (MPLMP_L) in this short-run scenario?
    Answer: 1 if L < 15, and 0 if L > 15. This is a Leontief (fixed-proportions) production function. With capital fixed at K=5, the production function becomes Q = min(L, 3*5) = min(L, 15). - If the firm hires labor L < 15, then L is the smaller value, so Q = L. In this range, each additional unit of labor adds exactly one unit to output. Thus, the marginal product of labor (MPLMP_L) is 1. - If the firm hires labor L > 15, then 15 is the smaller value, so Q = 15. In this range, adding more labor does not increase output at all, as the firm is constrained by its fixed capital. Thus, the marginal product of labor (MPLMP_L) is 0. - At L=15, the MP_L is undefined, but the function describes the behavior on either side.
  19. Consider a firm with production function Q=10LKQ = 10\sqrt{LK}. Currently, the firm uses 36 units of labor and 16 units of capital. If the firm increases labor to 49 units while keeping capital constant, what is the percentage change in the marginal product of capital?
    Answer: The marginal product of capital increases by approximately 16.7 percent First, rewrite the production function as Q = 10L^0.5K^0.5. The marginal product of capital is MPK = ∂Q/∂K = 5L^0.5K^(-0.5). Initially: MPK₁ = 5(36)0.5(36)^{0.5}(16)−0.5(16)^{-0.5} = 5(6)(1/4) = 7.5. After the change: MPK₂ = 5(49)0.5(49)^{0.5}(16)−0.5(16)^{-0.5} = 5(7)(1/4) = 8.75. Percentage change = (8.75 - 7.5)/7.5 × 100% = 16.7%. Choice B incorrectly calculates the percentage change in labor. Choice C gets the sign wrong. Choice D incorrectly assumes MPK is independent of L.
  20. A firm's production function is given by Q=20L0.6K0.4Q = 20L^{0.6}K^{0.4}, where L represents labor and K represents capital. If the firm currently uses 64 units of labor and 81 units of capital, what is the marginal product of labor at this input combination?
    Answer: 7.5 units of output per additional unit of labor The marginal product of labor (MPL) is the partial derivative of the production function with respect to L: MPL = ∂Q/∂L = 20(0.6)L−0.4L^{-0.4}K0.4K^{0.4} = 12L−0.412L^{-0.4}K0.4K^{0.4}. Substituting L = 64 and K = 81: MPL = 12(64)−0.4(64)^{-0.4}(81)0.4(81)^{0.4} = 12(1/4)(3) = 7.5. Choice B incorrectly uses the coefficient 12 without applying the exponents. Choice C doubles the correct answer. Choice D uses the total coefficient 20 plus an error in calculation.

Start this quiz